Step |
Hyp |
Ref |
Expression |
1 |
|
1re |
⊢ 1 ∈ ℝ |
2 |
|
chtval |
⊢ ( 1 ∈ ℝ → ( θ ‘ 1 ) = Σ 𝑝 ∈ ( ( 0 [,] 1 ) ∩ ℙ ) ( log ‘ 𝑝 ) ) |
3 |
1 2
|
ax-mp |
⊢ ( θ ‘ 1 ) = Σ 𝑝 ∈ ( ( 0 [,] 1 ) ∩ ℙ ) ( log ‘ 𝑝 ) |
4 |
|
ppisval |
⊢ ( 1 ∈ ℝ → ( ( 0 [,] 1 ) ∩ ℙ ) = ( ( 2 ... ( ⌊ ‘ 1 ) ) ∩ ℙ ) ) |
5 |
1 4
|
ax-mp |
⊢ ( ( 0 [,] 1 ) ∩ ℙ ) = ( ( 2 ... ( ⌊ ‘ 1 ) ) ∩ ℙ ) |
6 |
|
1z |
⊢ 1 ∈ ℤ |
7 |
|
flid |
⊢ ( 1 ∈ ℤ → ( ⌊ ‘ 1 ) = 1 ) |
8 |
6 7
|
ax-mp |
⊢ ( ⌊ ‘ 1 ) = 1 |
9 |
8
|
oveq2i |
⊢ ( 2 ... ( ⌊ ‘ 1 ) ) = ( 2 ... 1 ) |
10 |
|
1lt2 |
⊢ 1 < 2 |
11 |
|
2z |
⊢ 2 ∈ ℤ |
12 |
|
fzn |
⊢ ( ( 2 ∈ ℤ ∧ 1 ∈ ℤ ) → ( 1 < 2 ↔ ( 2 ... 1 ) = ∅ ) ) |
13 |
11 6 12
|
mp2an |
⊢ ( 1 < 2 ↔ ( 2 ... 1 ) = ∅ ) |
14 |
10 13
|
mpbi |
⊢ ( 2 ... 1 ) = ∅ |
15 |
9 14
|
eqtri |
⊢ ( 2 ... ( ⌊ ‘ 1 ) ) = ∅ |
16 |
15
|
ineq1i |
⊢ ( ( 2 ... ( ⌊ ‘ 1 ) ) ∩ ℙ ) = ( ∅ ∩ ℙ ) |
17 |
|
0in |
⊢ ( ∅ ∩ ℙ ) = ∅ |
18 |
5 16 17
|
3eqtri |
⊢ ( ( 0 [,] 1 ) ∩ ℙ ) = ∅ |
19 |
18
|
sumeq1i |
⊢ Σ 𝑝 ∈ ( ( 0 [,] 1 ) ∩ ℙ ) ( log ‘ 𝑝 ) = Σ 𝑝 ∈ ∅ ( log ‘ 𝑝 ) |
20 |
|
sum0 |
⊢ Σ 𝑝 ∈ ∅ ( log ‘ 𝑝 ) = 0 |
21 |
3 19 20
|
3eqtri |
⊢ ( θ ‘ 1 ) = 0 |